This paper presents a novel variational approach for optical flow estimation that integrates several key assumptions: brightness constancy, gradient constancy, and a discontinuity-preserving spatio-temporal smoothness constraint. The method avoids linearising the data terms to allow for large displacements and uses a consistent numerical scheme based on two nested fixed point iterations. The authors theoretically justify the use of warping, which has previously been used mainly experimentally, as a numerical approximation strategy. The method is shown to be robust to parameter variations and noise, and achieves significantly smaller angular errors compared to previous techniques. The paper also demonstrates that the method can be used with a multiscale approach to efficiently solve image correspondence problems. The authors further extend the brightness constancy assumption by incorporating a gradient constancy assumption, which improves robustness to brightness changes. Experimental results show that the method outperforms existing techniques in terms of accuracy and is efficient in computation. The method is evaluated on both synthetic and real-world data, and it is shown to produce accurate and realistic flow fields. The paper concludes that the combination of a continuous, rotationally invariant energy functional with a robust data term and a discontinuity-preserving spatio-temporal regulariser leads to superior performance in optical flow estimation. The authors also prove that the widely-used warping technique can be theoretically justified as a numerical approximation strategy that does not affect the continuous model.This paper presents a novel variational approach for optical flow estimation that integrates several key assumptions: brightness constancy, gradient constancy, and a discontinuity-preserving spatio-temporal smoothness constraint. The method avoids linearising the data terms to allow for large displacements and uses a consistent numerical scheme based on two nested fixed point iterations. The authors theoretically justify the use of warping, which has previously been used mainly experimentally, as a numerical approximation strategy. The method is shown to be robust to parameter variations and noise, and achieves significantly smaller angular errors compared to previous techniques. The paper also demonstrates that the method can be used with a multiscale approach to efficiently solve image correspondence problems. The authors further extend the brightness constancy assumption by incorporating a gradient constancy assumption, which improves robustness to brightness changes. Experimental results show that the method outperforms existing techniques in terms of accuracy and is efficient in computation. The method is evaluated on both synthetic and real-world data, and it is shown to produce accurate and realistic flow fields. The paper concludes that the combination of a continuous, rotationally invariant energy functional with a robust data term and a discontinuity-preserving spatio-temporal regulariser leads to superior performance in optical flow estimation. The authors also prove that the widely-used warping technique can be theoretically justified as a numerical approximation strategy that does not affect the continuous model.